Hydrodynamics of the N-BBM Process

Abstract

The Branching Brownian Motion (BBM) process consists of particles performing independent Brownian motions in \(\mathbb R\), and each particle creating a new one at rate 1 at its current position. The newborn particles’ increments and branchings are independent of the other particles. The N-BBM process starts with N particles and, at each branching time, the left-most particle is removed so that the total number of particles is N for all times. The N-BBM process has been originally proposed by Maillard, and belongs to a family of processes introduced by Brunet and Derrida. We fix a density \(\rho \) with a left boundary \(\sup \{r\in \mathbb R: \int _r^\infty \rho (x) d x=1\}>-\infty \), and let the initial particles’ positions be iid continuous random variables with density \(\rho \). We show that the empirical measure associated to the particle positions at a fixed time t converges to an absolutely continuous measure with density \(\psi (\cdot ,t)\) as \(N\rightarrow \infty \). The limit \(\psi \) is solution of a free boundary problem (FBP). Existence of solutions of this FBP was proved for finite time-intervals by Lee in 2016 and, after submitting this manuscript, Berestycki, Brunet and Penington completed the setting by proving global existence.